Consider the model [1]

$$y_n=X_n\beta_n+\epsilon_n$$ $$\beta_i|\sigma^2,v_i \sim \mathcal{N}(0,\sigma^2 v_i), i=1,\ldots,p$$ $$v_i \sim \beta^\prime(a,b)$$ $$\sigma^2 \sim \mathcal{IG}(c,d)$$

where $\beta^\prime$ is the beta prime distribution and $\mathcal{IG}$ the inverse gamma distribution.

In [1], the authors prove posterior consistency in the high-dimensionality regimes were $p\rightarrow \infty$ as $n\rightarrow \infty$. Is there a way to show posterior consistency for fixed/small $p$ as $n\rightarrow \infty$?

[1] Bai, R. and Ghosh, M., 2021. On the beta prime prior for scale parameters in high-dimensional bayesian regression models. Statistica Sinica, 31(2), pp.843-865.

  • $\begingroup$ @SextusEmpiricus, in arxiv.org/pdf/1807.06539.pdf the authors set $f$ to be a beta prime prior and prove posterior consistency in high dimension. My question is there a similar prove for fixed/small $p$ with the same prior and or different priors? $\endgroup$
    – MrDi
    Mar 6 at 21:54
  • $\begingroup$ @SextusEmpiricus, I think it is very familiar to anyone following the latest research of normal-scale mixture models and posterior consistency. $\endgroup$
    – MrDi
    Mar 6 at 22:11
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    $\begingroup$ @Ben, I have edited my question. $\endgroup$
    – MrDi
    Mar 8 at 6:59
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    $\begingroup$ @D.W., $f$ is the beta prime distribution. $\endgroup$
    – MrDi
    Mar 8 at 7:00
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    $\begingroup$ @SextusEmpiricus $\beta$ are the coefficients. $\endgroup$
    – MrDi
    Mar 8 at 7:02

1 Answer 1


Based on your reference I believe that you are estimating the vector $\boldsymbol{\beta}$ of size $p_n$ with a posterior distribution based on the observation of the vector $\mathbf{Y}$ of size $n$ in the model $$\mathbf{Y} = \mathbf{X} \boldsymbol{\beta} + \boldsymbol{\epsilon}$$ where $\mathbf{X}$ is a fixed regression matrix $n \times p_n$ matrix. The components $\beta_i$ from the vector $\boldsymbol{\beta}$ have a prior distribution that follows the model that you described.

If you have a fixed $p$ then consistency seems guaranteed by a Bayesian law of large numbers, or consistency of likelihood/posterior for an i.i.d sample (When do posteriors converge to a point mass?). Or maybe I am missing something?

Say we use some fixed $p$ while letting $n$ increase, then the Bayesian posterior density $f(\boldsymbol{\beta}|\mathbf{Y})$ will concentrate near the true $\boldsymbol{\beta}$ (if the prior is not zero).

The same is true when $p$ is not fixed, but still with a finite upper bound. We can consider all $p$'s below the bound together while letting $n \to \infty$ and if they are all individually consistent, then we will also have consistency when we change $p$ while letting $n$ increase.

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    $\begingroup$ I don't think this answers OP question. $\endgroup$
    – Isaac
    Mar 8 at 7:15
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    $\begingroup$ @Isaac I answered the question "Is there a way to show posterior consistency for fixed/small $p$" by explaining a way that it can be shown. Could you tell what is wrong with it? $\endgroup$ Mar 8 at 7:40
  • $\begingroup$ @SextusEmpiricus, could you elaborate more about this? How will the theorem in [1] change with fixed/small $p$? $\endgroup$
    – MrDi
    Mar 8 at 9:09
  • $\begingroup$ @MrDi for a fixed $p$ one can use the law of large numbers. When $p$ is not fixed then that approach becomes difficult because while we can have that for a larger $p$ the 'error' in the posterior may become larger. Then consistency is not obtained, if the increase in error due to increase in $p$ is faster than the decrease in error due to increase in $n$. $\endgroup$ Mar 8 at 9:20
  • $\begingroup$ @SextusEmpiricus, could you please reference some source where the law of large numbers is used to prove posterior consistency for the normal-scale mixture model showen the question? $\endgroup$
    – MrDi
    Mar 8 at 9:45

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